With respect to a rectangular cartesian coordinate system, three vectors are expressed as \vec{a} = 4\hat{i} - \hat{j} , \vec{b} = -3\hat{i} + 2\hat{j} and \vec{c} = -\hat{k} where \hat{i}, \hat{j}, \hat{k} are unit vectors, along the X, Y and Z-axis respectively. The unit vectors \hat{\Gamma} along the direction of sum of these vector is
\((a) \hat{\mathbf{r}} = \frac{1}{\sqrt{3}} (\hat{\mathbf{i}} + \hat{\mathbf{j}} - \hat{\mathbf{k}}) \hspace{1cm} (b) \hat{\mathbf{r}} = \frac{1}{\sqrt{2}} (\hat{\mathbf{i}} + \hat{\mathbf{j}} - \hat{\mathbf{k}}) (c) \hat{\mathbf{r}} = \frac{1}{3} (\hat{\mathbf{i}} - \hat{\mathbf{j}} + \hat{\mathbf{k}}) \hspace{1cm} (d) \hat{\mathbf{r}} = \frac{1}{\sqrt{2}} (\hat{\mathbf{i}} + \hat{\mathbf{j}} + \hat{\mathbf{k}})\)
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\vec{r} = \vec{a} + \vec{b} + \vec{c} = 4\hat{i} - \hat{j} - 3\hat{i} + 2\hat{j} - \hat{k} = \hat{i} + \hat{j} - \hat{k}
\(\hat{r} = \frac{\vec{r}}{|\vec{r}|} = \frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{1^2 + 1^2 + (-1)^2}} = \frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{3}}\)
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